The total number of 3 × 3 matrices A having enteries from the set 0 , 1 , 2 , 3 such that the sum of…

Let $A=\left[\begin{array}{lll}a& b& c\\ d& e& f\\ g& h& i\end{array}\right], A^T=\left[\begin{array}{lll}a& d& g\\ b& e& h\\ c& f& i\end{array}\right]$

Diagonal elements of $A{A}^T \text{is} a^2+b^2+c^2,{ d}^2+e^2+f^2,{ g}^2+b^2+c^2.$

Sum $=a^2+b^2+c^2+d^2+e^2+f^2+g^2+h^2+i^2=9$

$a,b,c,d,e,f,g,h,i\in \left\{0,1,2,3\right\}$

Then, the total no. of ways $=1+9+8\times 63+63\times 4$

$=766$